I am in 6th grade but to me, that doesn't matter because I have high hopes for myself. My question is; How many distinct letter arrangements can be made from the word LETTER with each T on each side?

(A) 6
(B) 12
(C) 24
(D) 120
(E) 720

I think it is 24 because if an example to one of the orders is TLEERT it would be 1x4x3x2x1x1 because the Ts are fixed so the 1s make sense and there is 4 letters left as an combination so would it be 4!.

Please answer

you want permutations of LEER, which is

4!/2! = 24/2 = 12

You have to divide by 2! because the two E's are indistinguishable.

LE1E2R looks just like LE2E1R

Since there are 2! ways to shuffle the E's, you need to divide the total by 2!. Read up on permutations with duplicates.

To find the number of distinct letter arrangements that can be made from the word "LETTER" with each "T" on each side, we can break down the problem into separate steps.

Step 1: We have to consider that there are two "T"s in the word "LETTER" that are fixed in their positions. The other four letters are "L", "E", "E", and "R". We need to find the number of arrangements for these four letters alone.

Step 2: For the four remaining letters ("L", "E", "E", and "R"), we can find the number of distinct arrangements by using the formula for the number of arrangements of objects with repetition.

The formula for the number of arrangements of objects with repetition is n!, divided by the product of the factorials of the repeated elements, where n is the total number of objects. In this case, n = 4.

So, using this formula:

Number of arrangements = 4! / (2! * 1! * 1!)

The factorials of the repeated elements "E" and "R" are 1, since they are not repeated more than once.

Simplifying:

Number of arrangements = (4 * 3 * 2 * 1) / (2 * 1 * 1) = 24 / 2 = 12

Therefore, there are 12 distinct letter arrangements that can be made from the word "LETTER" with each "T" on each side.

So the correct answer to the question is (B) 12.

Hope this helps!